Custom file output name format using incremental letters instead of numbers?
Well, this is awkward, I just figured it out. I had looked at an answer to a similar question here, but having no experience with scripting, I thought I wouldn't know how to adapt it to my needs. Turns out that after looking at some basic python tutorials, I managed to write a script that, while not very elegant, definitely works. Here's the script in question:

import bpy
letter = 'A'
def handler_set_filepath_based_on_frame(scene):
    if scene.frame_current == 1:
        letter = 'A'
    if scene.frame_current == 2:
        letter = 'B'       
    if scene.frame_current == 3:
        letter = 'C'
    if scene.frame_current == 4:
        letter = 'D'               
    if scene.frame_current == 5:
        letter = 'E'
    if scene.frame_current == 6:
        letter = 'F'               
    if scene.frame_current == 7:
        letter = 'G'  
    if scene.frame_current == 8:
        letter = 'H'        
    if scene.frame_current == 9:
        letter = 'I'  
    if scene.frame_current == 10:
        letter = 'J'  
    scene.render.filepath = ("/tmp/NAME%s0#" % letter)


bpy.app.handlers.frame_change_pre.append(handler_set_filepath_based_on_frame)


I'll likely will never need to render more than 10 frames or so. So manually defining a variable for each isn't that much of a hassle.

Here's the answer from which I based my script off of.
Changing filename format based on current animation frame?